Research question

• Are patterns of pandemic spread, its determinants, and effects of public health interventions are similar across pandemics in Switzerland?
• Estimate excess mortality for pandemics in 1890, 1918 and 2020 per district, age groups and sex
• Comparing of spatial pattern between the pandemics
• Investigate the determinants of spread in the context of different co-factors ( Urbanization, GIP per capita etc.)

Data

Methods

• Bayesian approach

• INLA (Bayesian inference for Latent Gaussian Models)

• Death data for \(130\) districts of Switzerland for all three pandemics (1890, 1918, 2020)

• Data \(Y\) is given as the total number of death in fixed areas and in each year

• Standard Poisson likelihood to model the counts \[ y_i \mid \eta_i\sim Po\left(E_i \exp(\eta_i)\right),\] where \(E_i\) is the ``population at risk’’ in region \(i\).
  
  

• classical disease mapping model; BYM model (Besag, York and Mollie proposed it)

• The log relative risk, \(\bf{\eta} = (\eta_1, \dots, \eta_n)^T\), is thus decomposed into \[\bf{\eta} = \mu + \bf{u} + \bf{v} \]

  • \(\mu\) is the overall intercept
  • \(\bf{u}\) is a random effect with spatial structure following the Besag model
  • \(\bf{v}\) represents a non-spatial overdispersion

• \({\bf u}\) is “besag” modelled spatially structured with smoothing parameter \(\kappa_u\).

• \({\bf v}\) is unstructured with precision parameter \(\kappa_v\), i.e. \(\bf{v} \sim \mathcal{N}(0, \kappa_v^{-1}I)\).

• The precision terms \(\kappa_v\) and \(\kappa_u\) are assigned the default gamma prior distributions of INLA \[ \begin{aligned} \kappa_u & \sim \textrm{Gamma}( \alpha_u, \beta_u ), \\ \kappa_v & \sim \textrm{Gamma}( \alpha_v, \beta_v ). \end{aligned} \]

• The default values are \(\alpha_u = \alpha_v = 1\) and \(\beta_u = \beta_v = 0.00005\).
  
  

• Year is modelled using independent and identically (iid) Gaussian prior distribution with \({N}(0, \tau_v^{-1})\)
  

• Calculation of expected values based on the mortality trend of the previous 4 years (1890 only 4 years possible)

 formula =
    death ~ 1 + offset(log(population)) +
    f(Region.struct, model="besag", graph="Bezirk_Inla", scale.model = TRUE)+
    f(Region, model='iid', constr = TRUE) +
    f(Year, model='iid', constr = TRUE) 
   


inla.mod = inla(formula,
                  data=reg_data,
                  family="Poisson",
                  verbose = TRUE,
                  control.compute=list(config = TRUE),
                  control.mode=list(restart=T),
                  num.threads = round(parallel::detectCores()*.8),
                  control.predictor=list(compute=T))

• 1000 samples from the posterior distribution
• Calculation of median and 95% CrI(Credible interval) of the 1000 samples
• Excess mortality = observed death counts – expected death counts

Results

Next steps, points to be discussed

1. Model choices? Now BMY (Besag)
2. Definition of priors and hyperparamter
3. Add further Co-factors such as:

• Urbanization
• Infant mortality rates as a proxy for health index
• Public health intervention for each district (canton)
• GDP per capita as proxy for SES
• Population density (population/km2)
• Proportion of children, 5–15 y (as school-age children are thought to drive influenza transmission)